We report the first detection of a mid-J isotopic CO line from an external galaxy. We detected the 13CO (6-5) line from the starburst nucleus of NGC 253. The line is suprisingly bright with an integrated intensity 7% of the 12CO (6-5) line, indicating optical depth in the 12CO line. Our LVG modeling shows that a single warm (T ~ 120 K), dense (n ~ 104 cm-3) component emits most of the 12CO and 13CO line emission from J = 2-1 through J = 7-6. The CO(1-0) line comes from an additional lower excitation envelope. About 60% of the total molecular gas mass within 70 pc of the nucleus is in the warm, dense component. We show that stellar far-UV photons or X-ray photons from a nuclear source are unlikely to be the primary sources of the gas heating. The most likely sources of heat are cosmic rays from the nuclear starburst or microturbulence within molecular clouds.